Let $X$ be an ordered set. A *down-set* (also called a *lower set* or an *order ideal*) of $X$ is a subset $D$ of $X$ such that for every $x, y \in D$, if $x \in D$ and $y \leq_X x$, then $y \in D$. The *down-set lattice* $\mathcal{O}(X)$ of $X$ is the set of all down-sets of $X$ ordered by inclusion.

Let $P$ and $Q$ be ordered sets, and let $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ be their respective down-set lattices. Are $P$ and $Q$ order-isomorphic if and only if $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic? If $P$ and $Q$ are order-isomorphic, then it's easy to show that $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic. And if $P$ and $Q$ are totally ordered, then $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic to the ideal completions of $P$ and $Q$ respectively, and it's again easy to show that if $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic, then $P$ and $Q$ are order-isomorphic. But what if $P$ and $Q$ are only partially ordered?